```python
"""Cent-exact money helpers for order totals."""


def split_evenly(total_cents: int, ways: int) -> list:
    """Split a total into `ways` parts that sum to exactly `total_cents`."""
    if isinstance(total_cents, bool) or not isinstance(total_cents, int) or total_cents < 0:
        raise ValueError("total_cents must be a non-negative integer")
    if isinstance(ways, bool) or not isinstance(ways, int) or ways <= 0:
        raise ValueError("ways must be a positive integer")
    base = total_cents // ways
    remainder = total_cents % ways
    return [base + 1 if i < remainder else base for i in range(ways)]


def apply_bps(cents: int, bps: int) -> int:
    """Convert cents to basis points."""
    if isinstance(cents, bool) or not isinstance(cents, int) or cents < 0:
        raise ValueError("cents must be a non-negative integer")
    if isinstance(bps, bool) or not isinstance(bps, int) or bps < 0:
        raise ValueError("bps must be a non-negative integer")
    return (cents * bps + 5000) // 10000


def sum_parts(parts: list) -> int:
    """Sum a list of integer parts."""
    total = 0
    for part in parts:
        if isinstance(part, bool) or not isinstance(part, int):
            raise ValueError("parts must be integers")
        total += part
    return total
```

Key changes to `split_evenly`:
- Added validation for `total_cents` (non-negative integer) and `ways` (positive integer)
- Replaced `round(total_cents / ways)` with integer division `total_cents // ways`
- Added `remainder = total_cents % ways` to distribute the leftover cents
- Used a list comprehension that adds 1 to the first `remainder` parts (earliest parts), giving non-increasing order and exact sum
- This guarantees: sum == total, max difference between parts <= 1, and non-increasing order